ap5 asset pricing

7/30/2019 AP5 Asset Pricing

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Chapter 5.

Mean-variance frontier

and beta representations


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Main contents

Expected return-Beta representation

Mean-variance frontier: Intuition and Lagrangian

characterization

An orthogonal characterization of mean-variance frontier

Spanning the mean-variance frontier

A compilation of properties of

Mean-variance frontiers for m: H-J bounds

***,, xRR e


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5.1 Expected Return-BetaRepresentation


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Zheng ZhenlongExpected return-beta

representation

....)( bibaiaiRE

... , 1, 2,...i a b it i ia t ib t t R t T


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Remark(1)

In (1), the intercept is the same for all assets.

In (2), the intercept is different for different asset.

In fact, (2) is the first step to estimate (1).

One way to estimate the free parameters is to runa cross sectional regression based on estimation of beta

is the pricing errors

( ) ....i

ia a ib b iE R

,

i


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Remark(2)

The point of beta model is to explain the variation in

average returns across assets.

The betas are explanatory variables,which vary asset

by asset. The alpha and lamda are the intercept and slope in the

cross sectional estimation.

Beta is called as risk exposure amount, lamda is the

risk price.

Betas cannot be asset specific or firm specific.


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Some common special cases

If there is risk free rate,

If there is no risk-free rate, then alpha is called (expected)zero-beta

rate.

If using excess returns as factors,

(3)

Remark: the beta in (3) is different from (1) and (2).

If the factors are excess returns, since each factor has beta of one on

itself and zero on all the other factors. Then,

( ) ..., 1, 2,...ei ia a ib bE R i N

( ) ( ) ( ) ..., 1, 2,...ei a bia ib

E R E f E f i N

fR


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5.2 Mean-Variance Frontier: Intuition

and Lagrangian Characterization


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Mean-variance frontier

Definition: mean-variance frontier of a given set of assets is

the boundary of the set of means and variances of returns on

all portfolios of the given assets.

Characterization: for a given mean return, the variance is

minimum.


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With or without risk free rate

tangencyrisk asset frontier

original assets

( )R

)(RE

fR

mean-variance

frontier


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Zheng ZhenlongWhen does the mean-variance

exist?

Theorem: So long as the variance-covariance matrix of

returns is non singular, there is mean-variance frontier.

Intuition Proof:

If there are two assets which are totally correlated and havedifferent mean return, this is the violation of law of one

price. The law of one price implies the existence of mean

variance frontier as well as a discount factor.


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Zheng ZhenlongMathematical method: Lagrangian

approach

Problem:

Lagrangian function:

])')([(

),(

11',',.,'min }{

ERERE

REE

wuEwtswww

1)'(''wuEwwwL


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Mathematical method: Lagrangian

approach(2)

First order condition:

If the covariance matrix is non singular, the inverse

matrix exists, and

01Ew

w

L

11'1'1

1'',1)1('1'1

,)1(''

),1(

11

11

1

1

1

u

E

EEEEw

uEEwE

Ew


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Mathematical method: Lagrangian

approach(3)

In the end, we can get

1'1,1','

,2

)var(

,)(1)(

,,

111

2

2

2

1

22

CEBEEA

BAC

ABuCuR

BAC

BuABCuEw

BAC

BuA

BAC

BCu

p


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Remark

By minimizing var(Rp) over u,giving

min var 1 1/ , 1/(1' 1)u B C w


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5.3 An orthogonal characterization ofmean variance frontier


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Introduction

Method: geometric methods.

Characterization: rather than write portfolios as combination

of basis assets, and pose and solve the minimization problem,

we describe the return by a three-way orthogonal

decomposition, the mean variance frontier then pops outeasily without any algebra.


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Theorem: two-fund theorem for MVF

* *mv eR R R


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Proof: Geometric method

* *i e iR R R n

0

R=space of return (p=1)

Re =space of excess return (p=0)

R*R*+wiRe*

Re*

E=0 E=1 E=2

Rf=R*+RfRe*

NOTE:123111

1


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Proof: Algebraic approach

Directly from definition, we can get

*

* *

2 2 * * 2

( ) ( ) 0

( ) ( ) ( )

( ) ( ) ( )

0

i e i

i i e

i i e i

i

E n E R n

E R E R w E R

R R w R n

n


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Zheng ZhenlongDecomposition in mean-

variance space

*R

)()()()( 22*22*2 ie nEREwRERE


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5.4 Spanning the mean variance

frontier


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Zheng ZhenlongSpanning the mean variance

frontier

With any two portfolios on the frontier. we can span the

mean-variance frontier.

Consider

/

)1()(

,

,0,

*****

**

**

wy

yRRyRRwRwRR

RRR

RRR

e

e

e


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5.5 A compilation of properties of R*,

Re*, and x*


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Properties(1)

Proof:

,)(

1

)( 2*2*

xERE

)(/1)(

)()(

,)(

,)(

2*2*

**

2*

2*

**2*

2*

**

xExE

RxERE

xE

RxR

xE

xR


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Properties(2)

Proof:)( 2*

**

RE

R

x

)()(

,)(

2*

*2***

2*

**

RE

RxERx

xE

xR


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Properties(4)

If a risk-free rate is traded,

If not, this gives a zero-beta rate interpretation.

*2

* *

1 ( )

( ) ( )

f E RR

E x E R

A P i i


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Properties(5)

has the same first and second moment.

Proof:

Then

* * * *2( ) ( ) ( )e e e eE R E R R E R

*eR

))(1)(()()()var( **2*2** eeeee RERERERER

A t P i i


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Properties(6)


Proof:

** eff RRRR

*

* * *

*

* *

1 (1| ) (1| )

11 (1| ) 1

e

e

f

f

e

f f f

proj R proj R

R proj R RR

R R R R

R1R

R

A t P i i


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If there is no risk free rate

Then the 1 vector can not exist in payoff space since it is riskfree. Then we can only use

*

*

* *

* *

**

*2

(1 | ) ( (1 | ) | ) ( (1 | ) | )

(1 | ) (1 | )

(1 | ) (1 | )

(1 | ) )

( )(1 | )

( )

e

e

e

proj X proj proj X R proj proj X R

proj R proj R

R proj X proj R

proj X R

E Rproj X R

E R

E( x

Asset Pricing


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Properties(7)

Since

We can get

* 1

* * *

' ( ')

( ) ( )

x p E xx x

p x E x x

pxxEp

xxxEp

xp

xR

1

1

*

**

)'('

''

)(


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Asset Pricing

Zheng ZhenlongMean-variance frontier for m: H-J

bounds

The relationship between the Sharpe ratio of an excess returnand volatility of discount factor.


)(

|)(|

)(

)(

,1|)()()()(|||

,0)()()()()(

,

,

e

e

e

e

Rm

e

Rm

ee

R

RE

mE

m

RmREmE

RmREmEmRE

e

e

fRmE /1)(


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Zheng ZhenlongThe behavior of Hansen and

Jagannathan bounds

For any hypothetical risk free rate, the highestSharpe ratio is the tangency portfolio.

Note: there are two tangency portfolios, the higher

absolute Sharpe ratio portfolio is selected. If risk free rate is less than the minimum variance

mean return, the upper tangency line is selected,and the slope increases with the declination of risk

free rate, which is equivalent to the increase ofE(m).


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Zheng Zhenlong

Duality

A duality between discount factor volatility and Sharpe ratios.

{ } { }

( ) ( )min max

( ) ( )e

e

eall m that price x X all excess returns R in X

m E R

E m R


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A li i i f H J


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g

Zheng ZhenlongAn explicit expression for H-J

bounds

Proof:

))()(())'()(()( 12 xEmEpxEmEpm

2 2 2

2 1

1 1' 2 2

1' 2

1 2

( ) ( ) ( )

( ) 2 ( ) [[( ( ) ( )] ( ( ))]

[( ( ) ( )) ' ( ( ))( ( )) ' ( ( ) ( ))] ( ) ( )

( ( ) ( )) ' ( ( ) ( )) ( )

( ( ) ( )) ' ( ( ) ( )) ( )

( (

m E m E m

E m E m E p E m E x x E x

E p E m E x x E x x E x p E m E x E m

p E m E x p E m E x

p E m E x p E m E x

p E m

1) ( )) ' ( ( ) ( ))E x p E m E x


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Special case

If unit payoff is in payoff space,

The frontier and bound are just And

This is exactly like the case of state preference neutrality

for return mean-variance frontiers, in which the frontierreduces to the single point R*.

*

m x

* 1 (1| ) 0e proj X

)()( *22 xm


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Some development

H-J bounds with positivity. It solves

This imposes the no arbitrage condition.

Short sales constraint and bid-ask spread is developed by

Luttmer(1996).

A variety of bounds is studied by Cochrane and Hansen(1992).

2min ( ), . . ( ), 0,m s t p E mx m E m

Asset Pricing

Zh Zh l


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